Motivated by successful application of ordinary derivative in various branches of science, the notion of derivation was introduced in rings and algebras long back. But the study of derivations in rings got impetus soon after Herstein and Posner simultaneously obtained some remarkable results particularly for prime rings in the year 1957. In recent years many well known algebraists such as Beidar, Bell, Bergen, Bre˘sar, Herstein, Martindale, Posner, Vukman and Ashraf ect. have made remarkable contributions to this area of study. The theory of derivations and automorphisms plays an important role not only in ring theory, but also in functional analysis, linear differential equations, concerning the question of innerness and outerness, for instance, the classical Noether-Skolem theorem yields the solution of the problem for finite dimensional central simple algebras. In the present thesis our objective is to study the results obtained by various authors concerning, derivations, semiderivation, (σ, τ )-derivation, Jordan (σ, τ )-derivations, left derivation, Jordan derivation and generalized Jordan derivations of prime and semiprime rings.